Learning General Sparse Additive Models from Point Queries in High Dimensions

Abstract

We consider the problem of learning a d-variate function f defined on the cube [−1, 1] d ⊂ R d , where the algorithm is assumed to have black box access to samples of f within this domain. Denote S r ⊂ [d] r ; r = 1, . . . , r 0 to be sets consisting of unknown r-wise interactions amongst the coordinate variables. We then focus on the setting where f has an additive structure, i.e., it can be represented aswhere each φ j ; j ∈ S r is at most r-variate for 1 ≤ r ≤ r 0 . We derive randomized algorithms that query f at carefully constructed set of points, and exactly recover each S r with high probability.In contrary to the previous work, our analysis does not rely on numerical approximation of derivatives by finite order differences.

High-dimensional nonlinear approximation by parametric manifolds in Hölder-Nikol'skii spaces of mixed smoothness

“…Preasymptotic estimation of high-dimensional problems were also treated in [16,17,18]. Related high-dimensional problems were studied in [22,30] based on ANOVA decomposition. The paper [11] has investigated dimension-dependent estimates of the approximation error for linear algorithms of sampling recovery on Smolyak grids of functions from the space with Hölder-Zygmund mixed smoothness.…”